Exercise 18.4
Page no 18.26
Question 1
Insert five G.M.'s between 16 and \frac{1}{4}.
Question 2
Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
Question 3
Insert three numbers between 1 and 256 so that the resulting sequence is a G.P.
Question 4
Insert six G.M.'s between \frac{8}{27} and -5 \frac{1}{16}.
Question 5
The sum of two numbers is six times their geometric mean. Show that the numbers are in the ratio 3+2 \sqrt{2}: 3-2 \sqrt{2}.
Question 6
If odd number of G.M.s are inserted between two given quantities a and b, show that the middle \mathrm{G}, \mathrm{M} .=\sqrt{a b}.
Question 7
(i) If A.M. and GM. of two positive numbers a and b are 10 and 8 respectively, find the numbers.
(ii) If A be the A.M. and G the GM. between two numbers, show that the numbers are A+\sqrt{A^{2}-G^{2}} and A-\sqrt{A^{2}-G^{2}}
Question 8
If the ratio of A.M. and G.M. between two numbers a and b is m: n, prove that a: b=m+\sqrt{m^{2}-n^{2}}: m-\sqrt{m^{2}-n^{2}}.
[H O T S]
Question 9
If one G.M., G and two A.M.'s p and q be inserted between two given quantities, prove that G^{2}=(2 p-q)(2 q-p).
|HOTS|
Question 10
If one A.M. A and two GM.'s p and q bc inserted between two given numbers, show that \frac{p^{2}}{q}+\frac{q^{2}}{p}=2 A.
[\mathrm{HOTS}]
Question 11
If a is the A.M. between b and c, b the G.M. between a and c, then show that \frac{1}{a}, \frac{1}{c}, \frac{1}{b} are in A.P.
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